The proposed research includes the development, analysis, and testing of mathematical models for the dynamics of populations, with the long-range goal of making more accurate population projections than have been possible in the past. In a sense, mathematical models summarize what we know at a given time of the exact way in which the various factors affecting population change, act and interact; at the moment, we seem to know very little, and much work remains to be done even to incorporate other demographic variables into models of closed populations, before considering the complexity added by including economic and behavioral variables or by considering open populations Several areas of research activity are proposed: most important seem the development of a satisfactory two-sex model, the study of the effects of perturbations over time in vital rates on population structure, and further investigation of random models for vital rates. although the primary emphasis in our work will be on human populations, most results should be more generally applicable to any closed biological population; in some areas, there is also considerable application of our models to other scholarly fields (e.g., models matching open jobs with unemployment persons). Recent developments in historical demography promise to yield time series of vital rates which will allow tests of our models, but many statistical problems arise in the analysis and interpretation of the data; we propose to attack these problems as the need arises.